Next, our model was used for TBG to calculate the TE properties at different twist angles. The twist-angle-resolved band structure of TBG used in this study is obtained from an openly available exact continuum model developed in Ref. [1]}, [2]}. The exact continuum model is based on \(k\cdot p\)  perturbation theory, combining the computational efficiency of tight-binding Hamiltonians with the twist-angle control offered by continuum models. In Fig. REF (a) and Fig. REF (b), we plot two representative examples of electronic band structures used in this study at twist angles \(\theta \)  = 1.01\(^\circ \)  and \(\theta \)  = 1.35\(^\circ \) , respectively, showing the flat bands near magic angle; the low energy bands open up with increasing twist angle (Fig. REF (b)) and eventually recover the Dirac cone at the K-point. The secondary band gap (E\(_g\)  shown in REF (b)) between the first and second subbands of electrons and holes influences carrier transport, especially at half-filling where the material exhibits correlated insulator behavior.[3]}
Due to the presence of flat bands and gaps, the DOS of TBG has sharp features that are affected by interactions with the lattice at temperatures of interest here. Collision broadening [4]}, [5]} from carrier scattering with phonons is implemented in this study by relating the scattering rate to the imaginary part of self energy \(\Gamma (k)=-(2/\hbar )\Im \Sigma (k)\)  and replacing the energy-conserving \(\delta \)  in Eqs. REF -REF  with a Lorentzian spectral function \((1/\pi ) \Gamma (k)/[(E(k)\pm \hbar \omega _{ph}-E(k^{\prime }))^2 + \Gamma (k)^2]\)  [6]}, [7]}. The DOS calculated after applying collision broadening is shown in Fig. REF (a) and Fig. REF (b) and compared with the unbroadened DOS to show that broadening partially smooths out some of the sharp features in the DOS. Acoustic phonon scattering in TBG is calculated with an acoustic deformation potential of D\(_{ac}\)  = 9 eV. This is in line with literature values that suggest D\(_{ac}\)  of TBG is 3 times higher than the value observed in SLG [8]}, [9]}, [10]}. D\(_{op}^\Gamma \)  = 10 eV nm\(^{-1}\)  with optical phonon energy of 0.150 meV (1213 cm\(^{-1}\) ) at \(\Gamma \) -point [11]} and D\(_{op}^K\)  = 20 eV nm\(^{-1}\)   [8]} with optical phonon energies of 0.148 meV (1197 cm\(^{-1}\) ) at K-point are used to calculate the inelastic phonon scattering rates.
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